| L(s) = 1 | + (−21.6 − 37.5i)2-s + (147. + 255. i)3-s + (85.1 − 147. i)4-s − 1.41e3·5-s + (6.38e3 − 1.10e4i)6-s + (−3.33e4 + 5.76e4i)7-s − 9.61e4·8-s + (4.51e4 − 7.82e4i)9-s + (3.05e4 + 5.29e4i)10-s + (4.22e5 + 7.31e5i)11-s + 5.01e4·12-s + (1.24e6 + 4.98e5i)13-s + 2.88e6·14-s + (−2.07e5 − 3.59e5i)15-s + (1.90e6 + 3.30e6i)16-s + (−3.62e6 + 6.28e6i)17-s + ⋯ |
| L(s) = 1 | + (−0.478 − 0.829i)2-s + (0.350 + 0.606i)3-s + (0.0415 − 0.0719i)4-s − 0.201·5-s + (0.335 − 0.580i)6-s + (−0.749 + 1.29i)7-s − 1.03·8-s + (0.254 − 0.441i)9-s + (0.0966 + 0.167i)10-s + (0.790 + 1.36i)11-s + 0.0581·12-s + (0.928 + 0.372i)13-s + 1.43·14-s + (−0.0706 − 0.122i)15-s + (0.454 + 0.788i)16-s + (−0.619 + 1.07i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.594 - 0.804i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.594 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.03790 + 0.523371i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03790 + 0.523371i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-1.24e6 - 4.98e5i)T \) |
| good | 2 | \( 1 + (21.6 + 37.5i)T + (-1.02e3 + 1.77e3i)T^{2} \) |
| 3 | \( 1 + (-147. - 255. i)T + (-8.85e4 + 1.53e5i)T^{2} \) |
| 5 | \( 1 + 1.41e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + (3.33e4 - 5.76e4i)T + (-9.88e8 - 1.71e9i)T^{2} \) |
| 11 | \( 1 + (-4.22e5 - 7.31e5i)T + (-1.42e11 + 2.47e11i)T^{2} \) |
| 17 | \( 1 + (3.62e6 - 6.28e6i)T + (-1.71e13 - 2.96e13i)T^{2} \) |
| 19 | \( 1 + (-1.15e6 + 2.00e6i)T + (-5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (-2.19e7 - 3.79e7i)T + (-4.76e14 + 8.25e14i)T^{2} \) |
| 29 | \( 1 + (7.31e7 + 1.26e8i)T + (-6.10e15 + 1.05e16i)T^{2} \) |
| 31 | \( 1 + 1.30e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + (-6.92e7 - 1.19e8i)T + (-8.89e16 + 1.54e17i)T^{2} \) |
| 41 | \( 1 + (-1.63e8 - 2.83e8i)T + (-2.75e17 + 4.76e17i)T^{2} \) |
| 43 | \( 1 + (-2.21e8 + 3.84e8i)T + (-4.64e17 - 8.04e17i)T^{2} \) |
| 47 | \( 1 + 4.33e8T + 2.47e18T^{2} \) |
| 53 | \( 1 + 3.27e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + (2.53e9 - 4.39e9i)T + (-1.50e19 - 2.61e19i)T^{2} \) |
| 61 | \( 1 + (-4.94e9 + 8.57e9i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (2.53e9 + 4.38e9i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 + (9.11e9 - 1.57e10i)T + (-1.15e20 - 2.00e20i)T^{2} \) |
| 73 | \( 1 - 1.80e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 2.16e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 3.45e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (-1.20e10 - 2.08e10i)T + (-1.38e21 + 2.40e21i)T^{2} \) |
| 97 | \( 1 + (1.58e10 - 2.74e10i)T + (-3.57e21 - 6.19e21i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.72085422329125318620415728126, −15.50748862234441769855499806773, −15.12020661808874281461675532343, −12.64390486043231285664725284040, −11.46347223522400997039083573484, −9.672820751340315266019156501398, −9.090525123298619231449101807690, −6.26049903453065707954351039092, −3.68008432064439435183066048669, −1.85512778536905528654304310410,
0.62285790865256883174688596329, 3.41705141586733527404952302296, 6.47589383177896521847345978483, 7.53536572881943417882770615150, 8.900523769907771041513538047070, 11.01656132188854921130518363111, 13.00769853929610509054530104781, 14.08830867632745407002504146973, 16.11353248749652891580979922051, 16.61589447549632031248492522867
